We introduce and discuss kinetic models for wealth distribution in a simple market economy, which are able to reproduce the salient features of the wealth distribution by including taxes to each trading process and redistributing the collected money among the population according to a given criterion. Our analysis gives a theoretical basis to some recent research that analyzed discrete simplified models for the exploitation of finite resources by interacting agents, where each agent receives a random fraction of the available resources. It is shown that in general the redistribution is able to modify the Pareto index, and that this modification can be quantified in terms of the redistribution operator.

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t → ∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L 1 -norm, as well as various Sobolev norms.

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We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and Htheorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

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We consider the spatially homogeneous Boltzmann equation for inelastic hardspheres (with constant restitution coefficient α ∈ (0, 1)) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter α ∈ (α0, 1), with some constructive α0 ∈ [0, 1). Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary solutions as the inelasticity parameter goes to 1. In order to achieve this we give an accurate spectral analysis of the associated linearized collision operator in the elastic limit. Several qualitative properties of this unique steady state Fα are also derived; in particular, we prove that Fα is bounded from above and from below by two explicit universal (i.e. independent of α) Maxwellian distributions.

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the L p -norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.